Use series to evaluate lim x->0 (x-ln(1+x)/x^2)?
I got 1/2. Could anyone please verify my answer. I am still very confused about series. Please show how you got your answer.
Thanks
The Taylor expansion method is used to approximate a function using a polynomial of infinite degree. It allows for the estimation of a function's values at a certain point, based on the values and derivatives of the function at a nearby point.
The Taylor expansion formula is derived using the Taylor series, which is an infinite sum of terms that represent the function's values and derivatives at a certain point. By truncating the series at a certain degree, we can obtain a polynomial that closely approximates the function.
The key assumptions for using the Taylor expansion method are that the function must be infinitely differentiable, meaning that all of its derivatives exist and are continuous. Additionally, the function must be centered at a specific point and the interval of interest must be small enough for the approximation to be accurate.
The Taylor expansion method can be used to find the value of a function at a point by truncating the Taylor series at a certain degree and evaluating the resulting polynomial at the desired point. The more terms included in the series, the more accurate the approximation will be.
Some limitations of using the Taylor expansion method include the fact that it can only be used for functions that are infinitely differentiable, which may not be the case for all functions. Additionally, the approximation may not be accurate if the interval chosen is too large or if the function has highly non-linear behavior. Finally, the calculation of higher order derivatives can be computationally intensive.